Quarter tone system

While 31edo is a system with fifth of tones, the quarter tone notations is handy and allow to name all the notes in a logical way. Because one tone is 5 degrees, between C and D (do and re) for example, we have the following notes:

Degree	Letter	Name	English full name
0	C	do	C
1	C+	do+	C half-sharp
2	C#	do#	C sharp
3	Db	re b	D flat
4	Dd	re d	D half-flat
5	D	re	D

Otherwise we can use double sharp and double flat:

Degree	Letter	Name	English full name
0	C	do	C
1	Dbb	do bb	D double flat
2	C#	do#	C sharp
3	Db	re b	D flat
4	Cx	re x	C double sharp
5	D	re	D

While using double sharp and double flat seem a bit confusing because it then alternates between C and D, it makes sense from a musical point of view. Indeed as far as harmonics and chords are concerned, using double sharp and double flat allow to have a way of writing chord that is consistent with traditional solfege.

Indeed, if we consider the subminor chord, and write it with D# for the second note and A# for the seventh harmonic, we get the following chords:

• C / D# / G / A#
• C# / Dx / G# / Ax
• Db / E / Ab / B
• D / E# / A / B#

So, as in 12-ET, we have the equation C# ~ Db and E# ~ F, in 31-ET, we have C# ≠ Db and E# ≠ F but we have:

• Cx = Dd
• C+ = Dbb
• E# = Fd
• E+ = Fb
• Ex = F+
• Ed = Fbb

It is not necessary to learn all by heart. Simply that there are 5 degrees in a tone, and 3 degrees in a diatonic semitone. So going from one note name to another name is always an odd difference of degrees. If the change is even, it can be written as sharp and flat.

An exception to using flat and sharp is the rast scale, where there is Ed and Bd, which are defined as such and not derived from other intervals.

Andrew Heathwaite system

Andrew Heathwaite proposes the following solfege system for singing the intervals of 31edo. Note that this is a subset of the syllables used for singing 41edo.

degrees of 31edo	syllable
0	do
1	di
2	ro
3	ra
4	ru
5	re
6	ri
7	ma
8	me
9	mu
10	mi
11	mo
12	fe
13	fa
14	fu
15	fi
16	se
17	su
18	so or sol
19	si
20	lo
21	le
22	lu
23	la
24	li
25	ta
26	te
27	tu
28	ti
29	to
30	da
31	do

See also: 17edo Solfege, 22edo Solfege, 29edo Solfege

Comments

For intervals that appear in the diatonic scale, the traditional solfege names are grandfathered in. While this makes it easier to learn the new syllables as extensions of the old ones -- if you are trained with the old ones to begin with -- it also makes for many irregularities.

The syllables do, re, mi, fa, so[l], la, ti have the same meaning as traditional major and perfect intervals. The names for minor intervals are also retained: ra, me, le, te, as well as the augmented fourth, fi, and diminished fifth, se. Some traditional names for chromatically-altered intervals appear here, but altered by a semisharp or semiflat, rather than a full sharp or flat: di for a semiaugmented unison, da for a semidiminished unison, ri for a semiaugmented second, fe for a semidiminished fourth, si for a semiaugmented fifth, and li for a semiaugmented sixth. The remaining syllables flesh out the septimal and undecimal intervals which are not represented in 12edo.

Note that there is little pattern to the traditional names.

Between do and fa, there is a somewhat consistent pattern in the syllables associated with each interval and the interval a perfect fifth above it. This is especially helpful for learning to sing tetrachordal scales and seventh chords. The irregularities between do and fa are grandfathered in from the traditional system and are easy to learn.

do => so[l]
di => si
ro => lo (lo is a "low" sixth)
ra => le (an irregularity from the traditional names)
ru => lu (the "u" vowel for undecimal intervals)
re => la (another irregularity grandfathered in; but notice the symmetry of the two irregularities)
ri => li
ma => ta
me => te (grandfathered in, but fits the pattern)
mu => tu (undecimal)
mi => ti (grandfather and fits)
mo => to
fe => da (breaks the pattern of vowels, but we do see the consonants change together)
fa => do (The pattern mostly breaks down here, and we are no longer within a tetrachord. However, there are a few fits, which are indicated below.)
fu => di
fi => ro
se => ra
su => ru (fits)
so[l] => re
si => ri (fits)
lo => ma
le => me (fits, and grandfathered)
lu => mu (fits)
la => mi (grandfathered)
li => mo
ta => fe
te => fa (grandfathered)
tu => fu (fits)
ti => fi (fits, and grandfathered)
to => se
da => su